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We give a necessary and sufficient geometric structural condition, which we call the a -Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities. The a -Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded C 1,a hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral n -varifold on a smooth (n+1) -dimensional Riemannian manifold satisfies the a -Structural Hypothesis for some a?(0,1/2) , its singular set is empty if n=6 , discrete if n=7 and has Hausdorff dimension =n-7 if n=8 ; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a -Structural Hypothesis for some a?(0,1/2) . The a -Structural Hypothesis on an n -varifold for any a?(0,1/2) is readily implied by either of the following two hypotheses: (i) the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing (n-1) -dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions.An optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

A general regularity theory for stable codimension 1 integral varifolds